Abstract It is known that if $$f:{{\mathbb R}}^2\rightarrow {\mathbb R}$$ f : R 2 → a polynomial in each variable, then f polynomial. We present generalizations of this fact, when $${{\mathbb R}}^...

A homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t → p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, an...

The paper introduces a new family of analytic bi-univalent functions that are injective and possess inverses, by employing q-analogue the derivative operator. Moreover, article establishes upper bounds Taylor–Maclaurin coefficients these functions, which can aid in approximating accuracy approximations using finite number terms. obtained Faber polynomial expansions. These apply to both initial ...

We compare the convergence behavior of best polynomial approximations and Legendre Chebyshev projections derive optimal rates for analytic differentiable functions in maximum norm. For functions, we show that approximation degree n is better than projection same by a factor $$n^{1/2}$$ . such as piecewise fractional smoothness, however, only some constant factors. Our results provide new insig...

In this article, by using Chebyshev’s polynomials and Chebyshev’s expansion, we obtain the best uniform polynomial approximation out of P2n to a class of rational functions of the form (ax2+c)-1 on any non symmetric interval [d,e]. Using the obtained approximation, we provide the best uniform polynomial approximation to a class of rational functions of the form (ax2+bx+c)-1 for both cases b2-4a...

ABSTRACT. Let P be a polynomial from R → R and D ∈ R. I will consider the properties of the class of ODEs Y ′ = P (y) ; Y (0) = D and their solutions. The solution space to these ODEs form a proper subspace of the analytic functions with ODEs. They have many interesting algebraic and topological properties. I will present efficient methods for generating the power series solutions to these poly...